

Preprint 80/2002
Closed Legendre geodesics in Sasaki manifolds
Knut Smoczyk
Contact the author: Please use for correspondence this email.
Submission date: 05. Sep. 2002
published in: New York journal of mathematics, 9 (2003), p. 23-47 (electronic)
Bibtex
MSC-Numbers: 53C44, 53C42
Keywords and phrases: legendre, curve shortening, geodesic, sasaki
Abstract:
If is a Legendre submanifold in a Sasaki manifold, then
the mean curvature flow does not preserve the Legendre condition. We define
a kind of mean curvature flow for Legendre submanifolds which slightly differs
from the standard one and then we prove that closed Legendre curves L
in a Sasaki space form M converge to closed Legendre geodesics, if
and
, where
denotes the
sectional curvature of the contact plane
and k,
are the curvature respectively the rotation number of L. If
, we obtain convergence of a subsequence to Legendre
curves with constant curvature. In case
and if the
Legendre angle
of the initial curve satisfies
, then we also prove convergence to a closed Legendre geodesic.